BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 1 Biomedical Imaging I Class 9 – Ultrasound...

BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 1

Biomedical Imaging IBiomedical Imaging I

Class 9 – Ultrasound Imaging

Doppler Ultrasonography;

Image Reconstruction

11/09/05


Doppler EffectDoppler Effect

Change in ultrasound frequency caused by motion of source (which can be a scatterer) and/or receiver relative to the background medium

Effect of receiver motion is different from that of source motion (Why?). Combining both effects gives:

If v = v’,

Rff

R

cff

c v

R

cff

c v

source velocity

observer velocity

cos1 , Doppler

cosd R

c vff ff shift

c v

2 coscosd

vff

c v

More generally,

cosR

cff

c v

θ

R


Clinical Application of Doppler EffectClinical Application of Doppler Effect

If v = v’,

But why would it ever be the case that source and detector both are moving in the same direction with the same speed?

2 coscosd

vff

c v

How about if the medium is moving past a stationary source and detector?

Limb

Artery

Ultrasound transmitter

Ultrasound receiver

see: C. Holcombe et al., “Blood flow in breast cancer and fibroadenoma estimated by colour Doppler ultrasonography,” British J. Surgery 82, 787-788 (1995).


Net Doppler ShiftNet Doppler Shift

Source Receive

r

f = wave frequency [s-1]

λ = wavelength [m]

c = wave (or phase, or propagation) velocity [m-s-1]

c = λf

c

λ


Net Doppler ShiftNet Doppler ShiftCase 1: source in motion relative to medium and receiver

v

λ´ = λ – v/f

= c/f – v/f = (c – v)/f

f’ = c/λ´ = [c/(c – v)]f

v = source speed [cm-s-1]

T = 1/f = wave period [s]

vT = v/f = distance source travels between emission of successive wavefronts (crests) [m]


Net Doppler ShiftNet Doppler ShiftCase 2: receiver in motion relative to medium and source

λ´ = λ

c´ = c + v

f’ = c´/λ = [(c + v)/c]f

v

v = receiver speed [m-s-1]

c’ = c + v = wave propagation speed in receiver’s frame of reference [m-s-1]


Net Doppler ShiftNet Doppler ShiftCase 3: source and detector in motion relative to each other and medium

vs

f’net = f’sf’r/f

= [c/(c – vs)][(c + vr)/c]f

= [(c + vr)/(c – vs)]f

fd = f’net – f = [(c + vr)/(c – vs) - 1]f

vr


Nonlinear Features of Ultrasound Wave Propagation Nonlinear Features of Ultrasound Wave Propagation

Pressure p can be expressed as a function of density η:

In combination with the fact that , we get

Note that if B 0, then c is a function of p. Wave crests (regions of compression) propagate faster than wave troughs (regions of rarefaction)!

Observable significance of this dependence is...?

0 0

0 0 0 0

2

0 0,

0 0

20 0 2

, ,

12

,

s

s s

p p A B

p pA B

2¶ ¶¶ ¶

condensation

0 0

2

,s

pc

¶¶

0 0

0 0,

2s

B cc

A p¶¶


Nonlinear Features of Ultrasound: Shock WavesNonlinear Features of Ultrasound: Shock Waves


Ultrasound Computed TomographyUltrasound Computed Tomography

Recommended supplemental reading:A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrasonic Imaging 4, 336-350 (1982).J. F. Greenleaf, “Computerized tomography with ultrasound,” Proceedings of the IEEE 71, 330-337 (1983).H. Schomberg, W. Beil, G. C. McKinnon, R. Proksa, and O. Tschendel, “Ultrasound computerized tomography,” Acta Electronica 26, 121-128 (1984).J. Ylitalo, J. Koivukangas, and J. Oksman, “Ultrasonic reflection mode computed tomography through a skullbone,” IEEE Transactions on Biomedical Engineering 37, 1059-1066 (1990).Kak and Slaney, Chapters 6 (Tomographic Imaging with Diffracting Sources) and 8 (Reflection Tomography)



Elementary Forms of Ultrasound CT Types

Ultrasonic Refractive Index Tomography

• Projection:

Ultrasonic Attenuation Tomography

• Projection:

• Three methods for estimating attenuation line integral:

– Energy-Ratio Method

– Division of Transforms Followed by Averaging Method

– Frequency-Shift Method

All of the foregoing are predicated on an assumption of negligible refraction/diffraction/scattering of ultrasound beams in the medium

1 , ,B

w d d wAn x y ds V T T T T

,B

Ax y ds



Kak and Slaney, pp. 153, 154



Kak and Slaney, p. 154

Photograph

Ultrasound Refractive Index

CT Image



Kak and Slaney, pp. 156-158

Ultrasonic Attenuation CT Images

E-ratio Method

f-shift Method

Division of Transforms

Method


Energy-Ratio MethodEnergy-Ratio Method

x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse, yw(t) = detected pulse for transmission through water

FT X(f), Y(f), Yw(f)

Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|.

E(fk) = energy (or power), at frequency fk, in H(f).

Consider any two specific frequencies, f1 and f2, for which E(f1) and E(f2) can be reliably and accurately determined. Then in principle:

1

2 1 2

1, ln

2

B

A

Ex y ds

ff E


Division of Transforms Followed by Averaging MethodDivision of Transforms Followed by Averaging Method

x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse, yw(t) = detected pulse for transmission through water

FT X(f), Y(f), Yw(f), Yw(f)

Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|.

HA(f) = -ln|H(f)| = -ln|Y(f)/Yw(f)|.

In principle:

2 2 1 1

2 2 1 12 1

1 1,

2 2

B ff ff

A AA ff ffx y ds H f df H f df

ff


Frequency-Shift MethodFrequency-Shift Method

x(t) = incident ultrasound pulse, y(t) = detected transmitted ultrasound pulse

FT X(f), Y(f), Yw(f)

f0 = frequency at which Yw(f) is maximal

fr = frequency at which Y(f) is maximal

σ2 = width of |Yw(f)|

In principle:

02,

2

Br

A

ffx y ds



Kak and Slaney, pp. 159-160

Ultrasound CT mammograph

y...

...compared with x-ray CT mammograms of the same

patient.


Diffraction Tomography with UltrasoundDiffraction Tomography with Ultrasound

What we can (attempt to) do when the “negligible refraction/diffraction/scattering” criterion mentioned earlier is violatedBased upon treating ultrasound propagation through medium as a wave phenomenon, not as a particle (i.e., ray) phenomenon

For homogeneous media, the Fourier Diffraction Theorem is analogous to the central-slice theorem of x-ray CTHeterogeneous media are treated as (we hope) small perturbations of a homogeneous medium, to which an assumption such as the Born approximation or Rytov approximation can be applied


Fourier Diffraction Theorem IFourier Diffraction Theorem I

Arc radius is ultrasound

frequency, or wavenumber

Kak and Slaney, pp. 219


Fourier Diffraction Theorem IIFourier Diffraction Theorem II

Kak and Slaney, pp. 228, 229

Arc radius dependenceon wavenumber

Tomographic measurements fill up

Fourier space

BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 1 Biomedical Imaging I Class 9 – Ultrasound...

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